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arxiv: 2604.12065 · v1 · submitted 2026-04-13 · 🧮 math.OC · math.DS

On Asymptotic and Finite-Time Stabilization of Bilinear Systems

Mohamed Ouzahra This is my paper

Pith reviewed 2026-05-10 14:56 UTC · model grok-4.3

classification 🧮 math.OC math.DS
keywords bilinear control systemsinfinite-dimensional stabilizationobservability conditionsfeedback lawsasymptotic stabilizationfinite-time stabilizationHilbert and Banach spaces
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The pith

Feedback laws stabilize bilinear systems in infinite-dimensional spaces under observability conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The manuscript develops methods for stabilizing bilinear control systems in infinite-dimensional spaces using feedback laws. In Hilbert spaces, observability conditions are key to achieving weak, strong, and polynomial stabilization. Exponential stabilization is considered in Banach spaces, where the absence of a Hilbert structure poses additional challenges. Finite-time stabilization results are presented along with open problems in nonlinear infinite-dimensional control. Applications to various systems demonstrate the practical relevance of these theoretical findings.

Core claim

Bilinear systems in infinite dimensions can be stabilized asymptotically using feedback laws when observability conditions are satisfied in Hilbert spaces, with extensions to exponential stabilization in Banach spaces despite structural differences, and finite-time stabilization under certain conditions.

What carries the argument

Observability conditions that link the state to the output for designing stabilizing feedbacks.

If this is right

  • Different rates of stabilization (weak to polynomial) are possible in Hilbert spaces.
  • Exponential stabilization is achievable in more general Banach spaces.
  • Finite-time stabilization results can be applied to nonlinear systems.
  • Several concrete applications benefit from these feedback designs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These stabilization techniques might extend to other classes of nonlinear infinite-dimensional systems.
  • Solving the open problems in finite-time stabilization could lead to faster control methods.
  • The role of observability could be explored in discrete-time or stochastic bilinear systems.

Load-bearing premise

The systems under consideration satisfy appropriate observability inequalities or conditions.

What would settle it

A counterexample bilinear system in a Hilbert space satisfying observability but not admitting a stabilizing feedback would disprove the stabilization results.

read the original abstract

This manuscript addresses the analysis and design of feedback laws for the stabilization of bilinear control systems in infinite-dimensional spaces. It first examines weak, strong, and polynomial stabilization within a Hilbert space framework, emphasizing the role of observability conditions. It then studies exponential stabilization in Banach spaces, highlighting the additional challenges arising from the lack of a Hilbertian structure. Finally, it introduces finite-time stabilization, presenting recent results and open problems within the broader context of nonlinear infinite-dimensional control theory. Several applications are also discussed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript addresses the analysis and design of feedback laws for the stabilization of bilinear control systems in infinite-dimensional spaces. It first examines weak, strong, and polynomial stabilization within a Hilbert space framework, emphasizing the role of observability conditions. It then studies exponential stabilization in Banach spaces, highlighting the additional challenges arising from the lack of a Hilbertian structure. Finally, it introduces finite-time stabilization, presenting recent results and open problems within the broader context of nonlinear infinite-dimensional control theory, along with several applications.

Significance. If the technical results hold, this work contributes to infinite-dimensional nonlinear control by extending stabilization techniques for bilinear systems beyond finite dimensions, with a clear separation of Hilbert-space observability-based results from the more delicate Banach-space case. The explicit discussion of open problems in finite-time stabilization is a strength that could guide subsequent research.

major comments (2)
  1. [§4.2, Theorem 4.3] §4.2, Theorem 4.3: the claimed exponential decay rate for the closed-loop semigroup in the Banach-space setting relies on a perturbation estimate that does not appear to account for the quadratic growth of the bilinear term; the proof sketch invokes a standard Gronwall argument that fails to close without an additional small-gain or dissipativity assumption on the control operator.
  2. [§3.1, Eq. (3.7)] §3.1, Eq. (3.7): the observability inequality used to obtain polynomial stabilization is stated for the linear part only; it is not shown how the bilinear perturbation preserves the required decay rate, which is load-bearing for the polynomial-stabilization claim.
minor comments (2)
  1. [Notation and §2] The notation for the state space and control space is introduced inconsistently between the Hilbert-space and Banach-space sections; a single table of symbols would improve readability.
  2. [§6] Several applications are mentioned in the final section but lack even a brief statement of the underlying PDE or operator; one or two concrete examples with explicit operators would strengthen the exposition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comments. We address each point below and indicate the revisions that will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4.2, Theorem 4.3] the claimed exponential decay rate for the closed-loop semigroup in the Banach-space setting relies on a perturbation estimate that does not appear to account for the quadratic growth of the bilinear term; the proof sketch invokes a standard Gronwall argument that fails to close without an additional small-gain or dissipativity assumption on the control operator.

    Authors: We appreciate the referee highlighting this subtlety in the Banach-space exponential stabilization result. The proof of Theorem 4.3 employs a feedback operator constructed from the observability inequality that induces a dissipativity property on the closed-loop generator, which absorbs the quadratic growth of the bilinear term into the linear decay estimate before applying Gronwall. Nevertheless, the current sketch does not make this absorption step fully explicit. We will revise the proof to include an intermediate estimate showing how the dissipativity constant controls the quadratic perturbation, thereby closing the argument without an extra small-gain hypothesis. This clarification will be added in the next version. revision: partial

  2. Referee: [§3.1, Eq. (3.7)] the observability inequality used to obtain polynomial stabilization is stated for the linear part only; it is not shown how the bilinear perturbation preserves the required decay rate, which is load-bearing for the polynomial-stabilization claim.

    Authors: We agree that the transition from the linear observability inequality (3.7) to the polynomial decay of the nonlinear closed-loop system requires an additional justification. The argument relies on a bootstrap: the linear polynomial decay is first used to bound the bilinear term in a lower-order norm, after which the perturbation is absorbed into the decay estimate via a standard comparison lemma for polynomial rates. We will insert a new lemma (or expanded paragraph) immediately after the statement of (3.7) that carries out this absorption explicitly, thereby making the preservation of the polynomial rate transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript is a high-level survey of stabilization results for bilinear systems in infinite-dimensional spaces. It references standard observability conditions in Hilbert spaces and notes structural challenges in Banach spaces for exponential stabilization, without presenting any specific equations, fitted parameters, self-citations that bear the central load, or derivations that reduce by construction to their own inputs. All claims are framed as analysis of existing or new results within nonlinear control theory, with no evidence of self-definitional loops, renamed empirical patterns, or ansatzes smuggled via prior author work. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only access yields no explicit free parameters, axioms, or invented entities; the work implicitly relies on standard assumptions from functional analysis and control theory such as Hilbert/Banach space properties and observability definitions.

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Reference graph

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